Mathematical Modeling and Applications in Nonlinear Dynamics by Albert C.J. Luo & Hüseyin Merdan

Author:Albert C.J. Luo & Hüseyin Merdan
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(4.3)

where q i : [s i , t i+1] × X → X is a T-periodic continuous function for all with q i+m  = q i , and is another T-periodic continuous function for all with g i+m  = g i . To achieve our aim, we have to show the continuity and compactness of the corresponding Poincaré map P: X → X of (4.3) given by (4.7). Then we establish new sufficient conditions on the existence of periodic mild solutions when PC-mild solutions are ultimate bounded (see Theorems 2 and 3). Furthermore, a global asymptotic stability result of periodic solutions is presented in Theorem 4. These results are applied in Sects. 4.4 and 4.5 for the random cases (4.1) and (4.2). The final section, Sect. 4.6, is devoted to concrete examples to illustrate the theory.

This chapter is a continuation of our recent related papers [11, 29]. However, we note that it seems that we are the first to study the above evolution equations with random noninstantaneous impulses, which is the main novelty of this work. It is interesting that fractals are studied similarly in [16] as in our chapter for an economic, random, discrete-time, two-sector optimal growth model in which the production of the homogeneous consumption good uses Cobb–Douglas technology.

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